PALINDROMIC SUMS OF PROPER DIVISORS
Paul Pollack
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
[email protected]
Received: , Revised: , Accepted: , Published:
Abstract
Fix an integer g ≥ 2. A natural number n is called a palindrome in base g if its base g expansion
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reads the same forwards and backwards. Let s(n) = d|n, d<n d be the sum-of-proper-divisors
function. We show that for almost all (that is, asymptotically 100% of) natural numbers n, s(n)
is not a palindrome in base g. We also show how to reach the same conclusion for several other
commonly occurring arithmetic functions.
1. Introduction
Fix an integer g ≥ 2. We say that a natural number n is a palindrome in base g (or g-palindromic)
if its base g expansion reads the same forwards and backwards. In other words, if we write
n = a0 + a1 g + · · · + at g t ,
where each ai ∈ {0, 1, . . . , g − 1} and at > 0,
then ai = at−i for i = 0, 1, 2, . . . , t. A simple counting argument shows that the number of integers
in [1, x] that are palindromic in base g has order of magnitude x1/2 for all x ≥ 1.
Given a naturally occurring integer sequence, one might ask about the frequency with which
this sequence intersects the set of palindromes. This has been studied for the sequence of primes
[1, 5], linear recurrence sequences [11, 4], and the sequences of kth powers for fixed k = 2, 3, . . .
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[3]. Here we investigate this problem for the sequence {s(n)}∞
n=1 , where s(n) :=
d|n, d<n d is the
sum-of-proper-divisors function. Our main theorem is that s(n) is g-palindromic only for a density
zero set of natural numbers n. In fact, we prove a bit more.
Definition. Let k be a positive integer. We say that the positive integer n is k-nearly-palindromic
in base g if either n < g 2k , or n ≥ g 2k and the first k digits of n coincide with the reversal of the
last k digits of n.
It is clear that every palindrome is k-nearly-palindromic for each k = 1, 2, 3, . . . .
Theorem 1. Fix g ≥ 2. Let k be an integer with k ≥ 2. The upper density of those n for which
s(n) is k-nearly-palindromic is Og (1/ log k).
In the opposite direction from Theorem 1, palindromic values of s(n) are at least as frequent
as the primes, for the trivial reason that s(n) = 1 for all prime values of n. However, we do
not know how to prove that in each base g, the function s(n) assumes infinitely many distinct
palindromic values; for instance, we do not know how to do this when g = 10. (This question is
uninteresting if, e.g., g is prime, since then s(g k ) is always palindromic.) This would follow from
the conjecture that all large even n can be written as a sum p + q, with p and q distinct primes (a
slight strengthening of Goldbach), since we can then arrange for s(pq) = p + q + 1 to coincide with
any large odd number. But the existing results on the Goldbach conjecture seem, even under the
Generalized Riemann Hypothesis, to be too weak to say anything about our problem.
One might wonder why we concentrate on the particular function s(n). It turns out that for
most of the other commonly occurring arithmetic functions, the situation is much simpler. In §3,
we show that for each function f ∈ {σ, ϕ, λ, d, ω, Ω}, the set of n for which f (n) is palindromic is
a set of density zero.
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Notation and conventions
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We continue to use σ(n) = d|n d for the usual sum-of-divisors function, ϕ(n) = #(Z/nZ)× for the
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Euler function, λ(n) for the Carmichael λ-function giving the exponent of (Z/nZ)× , d(n) = d|n 1
for the number-of-divisors function, and Ω and ω for the functions counting the number of prime
divisors, with and without multiplicities, respectively. We use O and o-notation, as well as the
symbols , , and , with their usual meanings. Dependence of implied constants is indicated
with subscripts. For x > 0, we let log1 x = max{1, log x}, and we let logk denote the kth iterate
of log1 . Note that with this convention, logk x ≥ 1 for every x > 0.
2. Palindromic values of the sum-of-proper-divisors function
For each real u, let D(u) = {n ∈ N : s(n) ≤ un}. In 1933, Davenport showed that the sets
D(u) possess an asymptotic density for every real u [6]. Calling this density D(u), he proved that
D(u) is continuous everywhere and that limu→∞ D(u) = 1. The following result is an analogue of
Davenport’s theorem for arithmetic progressions.
Lemma 2. Let a and q be integers with q > 0. For each real u, let Da,q (u) = {n ≡ a mod q :
s(n) ≤ un}. Then for all u,
# (Da,q ∩ [1, x])
Da,q (u) := lim
x→∞
x/q
exists, and Da,q is a continuous function of u with limu→∞ Da,q (u) = 1.
Proof. While the discussion so far has been phrased in terms of s(n)/n, everything is easily translated to be about the function σ(n)/n, since σ(n)/n = 1 + s(n)/n. In particular, Davenport’s
theorem may be read as asserting the existence of a continuous limiting distribution for σ(n)/n,
while Lemma 2 amounts to the claim that σ(n)/n has a continuous distribution function when n
is restricted to the progression a (mod q).
For a wide class of arithmetic functions, H. N. Shapiro showed that the existence of a distribution
function relative to N implies the existence of a distribution function relative to each arithmetic
progression a (mod q) [14, Theorem 5.2]. Shapiro’s result applies in particular to σ(n)/n. The
continuity of the resulting distribution function follows immediately from the continuity of Davenport’s function D(u).
A priori, one might expect q distinct distribution functions Da,q corresponding to the q different
choices for a (mod q). In fact, there is quite a bit of redundancy.
Lemma 3. Let a, b, and q be integers with q > 0. If gcd(a, q) = gcd(b, q), then Da,q = Db,q .
Essential to the proof of Lemma 3 is the following result from probability, which is one concrete
embodiment of the method of moments. See, for example, the textbook of Billingsley [2, Theorems
30.1 and 30.2, pp. 406–408].
Lemma 4. Let F1 , F2 , F3 , . . . be a sequence of distribution functions. Suppose that each Fi corresponds to a probability measure on the real line concentrated on [0, 1]. For each k = 1, 2, 3, . . . ,
assume that
Z
µk := lim
uk dFj (u)
j→∞
exists. Then there is a unique distribution function F possessing the µk as its moments, and Fn
converges weakly to F as n → ∞.
Proof of Lemma 3. Rather than s(n)/n or σ(n)/n, it is convenient in this proof to work instead
with the function n/σ(n), which is universally bounded between 0 and 1. Thus, for each a and q,
we define
1
n
D̃a,q (u) = lim
#{n ≤ x : n ≡ a (mod q) and
≤ u}.
x→∞ x/q
σ(n)
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A quick calculation shows that for u ≥ 0, Da,q (u) = 1 − D̃a,q ((u + 1)−1 ). So to prove the lemma,
it suffices to show that D̃a,q = D̃b,q if gcd(a, q) = gcd(b, q).
By Lemma 4, it is enough to prove that for every k ∈ N, the kth moments of Da,q and Db,q
agree. Equivalently, it suffices to show that for each such k,


k
k
X
X

n
n
1
 = 0.
lim 
−
x→∞ x 
σ(n)
σ(n) 
n≤x
n≡a (mod q)
n≤x
n≡b (mod q)
Define an arithmetic function h so that (n/σ(n))k =
X
n≤x
n≡a (mod q)
n
σ(n)
k
−
X
n≤x
n≡b (mod q)
n
σ(n)
k
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d|n
h(d) for each n. Then




=
h(d) 

d≤x

X
X
1−
n≤x
n≡a (mod q)
d|n
X
n≤x
n≡b (mod q)
d|n




1 .


(1)
We are assuming that gcd(a, q) = gcd(b, q), so that a and b share the same set of common divisors
with q. Hence, gcd(d, q) divides a if and only if gcd(d, q) divides b. It follows that the parenthesized
difference of sums in (1) is either a difference of empty sums, or is a difference of two sums both of
which count numbers in [1, x] belonging to a prescribed congruence class modulo lcm[q, d]. Hence,
this difference of sums is always bounded by 1 in absolute value. Thus, the proof of the lemma
will be completed if can show that
X
|h(d)| = o(x),
d≤x
as x → ∞. For primes p and positive integers e, we have
h(pe ) = (pe /σ(pe ))k − (pe−1 /σ(pe−1 ))k ,
k
1
which makes clear that |h(pe )| ≤ 1. Moreover, h(p) = 1 − p+1
− 1; from the mean value
theorem applied to t 7→ (1 − t)k , we deduce that
|h(p)| ≤
k
k
<
p+1
p
for each prime p. Since every d can be decomposed as the product of a squarefree number d1 and
a coprime squarefull number d2 ,
X
X
X
|h(d1 )|
|h(d2 )|
|h(d)| ≤
d2 ≤x
d2 squarefull
d1 ≤x
d1 squarefree
d≤x
X
≤
d1 ≤x
d1 squarefree
x
1/2
Y
p≤x
k ω(d1 )
d1
k
1+
p
X
1
d2 ≤x
d2 squarefull
k x1/2 (log x)k .
This is certainly o(x), and so the proof is complete.
Lemma 5. Let W be a fixed positive integer. Then W | σ(n) for all n outside of a set of asymptotic
density zero.
Proof. Watson, investigating unpublished claims of Ramanujan, showed that the number of n ≤ x
for which W - σ(n) is O(x/(log x)1/ϕ(W ) ) for x ≥ 2 [15, Hauptsatz 2]. The implied constant in
Watson’s result may depend on W . Watson’s theorem is enough for our present purposes, but
we note that from [13, Theorem 2], one can deduce that the same O-estimate holds uniformly in
W.
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If I is a bounded interval of the real line and F is a continuous distribution function, we let
F (I) = F (b) − F (a), where a < b are the endpoints of I. The next lemma is a weak form of a
theorem of Erdős [9, Theorem, p. 60].
Lemma 6. Let I be a bounded interval of the real line of length |I| > 0. Then for Davenport’s
distribution function D, we have D(I) 1/ log(2 + |I|−1 ).
We can now prove our main result.
Proof of Theorem 1. Let Dk be the set of positive integers n for which s(n) is k-nearly-palindromic
and let Dk (x) := Dk ∩ [1, x). To prove the theorem, it suffices to show that
lim sup
m→∞
1
#Dk (g m )
g
.
gm
log k
We proceed to estimate the cardinality of Dk (g m ) for large m. When counting elements n ∈
Dk (g m ), we may assume all of the following:
(i) s(n)/n > 1/k,
(ii) s(n)/n < k,
(iii) n > g m / log k,
(iv) σ(n) ≡ 0 (mod g k ).
(v) gcd(n, g k ) ≤ (g log (2k))2ω(g) .
Indeed, taking I = [0, 1/k] in Lemma 6 shows that the number of n ≤ g m violating (i) is
O(g m / log k) for large m. (Throughout the proof, the notion of ‘large’ may depend on both g
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and k.) Since n≤gm s(n)/n ≤ n≤gm σ(n)/n ≤ g m d≤gm 1/d2 < 2g m , there are only O(g m /k)
values of n ≤ g m violating (ii). That we can assume (iii) is trivial, and that we can assume (iv) is
immediate from Lemma 5. Now we turn to (v). If (v) fails for n, then
Y
pe ≥ gcd(n, g k ) > (g log (2k))2ω(g) ,
pe kn
p|g
and so n is divisible by some prime power pe > (g log (2k))2 , where p | g. Clearly, e > 1. In
particular, n has a squarefull divisor exceeding (g log (2k))2 ; but the number of such n ≤ g m is
Og (g m / log k). Thus, (v) is safe to assume.
For later use, we record that (i) and (iii) imply
gm
log k
n
≥ g m−` , where ` := 2
.
s(n) > >
k
k log k
log g
Now let
A = the integer formed by the first k digits of n,
B = the integer formed by the last k digits of s(n).
Observe that since
n = σ(n) − s(n) ≡ −s(n) ≡ −B
(mod g k ),
the last k digits of n are determined by B.
For large enough values of m, all n under consideration have s(n) > g m−` > g 2k . Since s(n)
is k-nearly-palindromic, the first k digits of s(n) are formed by reversing the digits of B. (In
particular, the last digit of B is not zero.) Let B̃ be the integer formed by reversing the digits of
B. Then
A · g a ≤ n < (A + 1)g a , B̃ · g a+b ≤ s(n) < (B̃ + 1) · g a+b
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for certain integers a and b, and
B̃ b
s(n)
B̃ b
1
1
.
g 1−
≤
≤ g 1+
A
A+1
n
A
B̃
Since both B̃ and A have k digits in base g, we see that B̃/A g 1. Now from (i) and (ii),
b = Og (log k). Thus,
"
#
s(n)
B̃ b
−k B̃ b
−k
∈
g − C1 kg , g + C2 kg
(2)
n
A
A
for certain constants C1 and C2 depending only on g. Note that n has a+k digits, and so a ≤ m−k,
since n < g m . Since n > g m / log k ≥ g m−` , we also have a > m − k − `.
For fixed A, B, a, and b, we estimate the number of n ∈ [A · g a , (A + 1)g a ) having n ≡ −B
(mod g k ) and satisfying (2). Let I denote the interval appearing on the right-hand side of (2).
Then the number of these n is at most
(A + 1) · g a
D−B,gk (I)
D−B,gk (I)
− A · ga
+ o(g r ) = D−B,gk (I)g a−k + o(g r ),
k
g
gk
(3)
where the o-estimates are valid as m → ∞. Since n ≡ −B (mod g k ), we have
d : = gcd(−B, g k )
= gcd(n, g k ) ≤ (g log (2k))2ω(g) ,
(4)
by (v). From Lemma 3,
D−B,gk (I) · #{M mod g k : gcd(M, g k ) = d} =
X
DM,gk (I)
k
M mod g
gcd(M,g k )=d
= g k · lim
x→∞
1
#{n ≤ x : gcd(n, g k ) = d and s(n)/n ∈ I} ≤ g k D(I).
x
Thus,
gk
D(I)
#{M mod g k : gcd(M, g k ) = d}
gk
D(I).
=
ϕ(g k /d)
D−B,gk (I) ≤
k
Now ϕ( gd ) =
gk
d
Q
p|g k /d (1 − 1/p)
g
gk
d ;
using this and our upper bound (4) on d, we obtain that
D−B,gk (I) g (g log (2k))2ω(g) D(I) g (log (2k))2ω(g) D(I).
Since |I| g kg −k , Lemma 6 gives that D(I) g k −1 . Hence,
D−B,gk (I) g
(log (2k))2ω(g)
.
k
Using this estimate in (3), and summing over a ∈ {m − k − ` + 1, m − k − ` + 2, . . . , m − k}, we
see that the number of n that arise from fixed choices of A, B, and b is
g g m−2k
(log (2k))2ω(g)
,
k
up to an error term that is o(g m ) as m → ∞.
Finally, we sum over the O(g k ) possibilities for A, the O(g k ) possibilities for B, and the Og (log k)
possibilities for b. We conclude that
lim sup
m→∞
as desired.
#Dk (g m )
1
(log (2k))2ω(g)+1
1
g
+
g
,
m
g
log k
k
log k
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3. Other arithmetic functions
3.1. The functions ω, Ω, and d
We need two results concerning the distribution of the number of prime factors of n for typical
values of n.
p
Lemma 7. Let K ≥ 1 and let x ≥ 1. The number of n ≤ x for which |ω(n) − log2 x| > K log2 x
is O(x/K 2 ). The same estimate holds with ω replaced by Ω.
Proof. This follows immediately from the theorem of Turán that for each of f = ω and f = Ω, we
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have n≤x (f (n) − log2 x)2 = O(x log2 x). See, for example, [7, pp. 94–97].
Lemma 8. Let x ≥ 1. Then
max #{n ≤ x : ω(n) = t} p
t
x
,
log2 x
where the maximum is taken over all nonnegative integers t. The same theorem holds with ω
replaced by Ω.
Proof. When t = 0, there is precisely one integer n ≤ x with ω(n) = t, namely n = 1. Suppose
now that t ≥ 1. According to a theorem of Hardy and Ramanujan [10, Lemma A],
t−1
#{n ≤ x : ω(n) = t} x (log2 x + c)
(log x)
(t − 1)!
,
(5)
for a certain absolute positive constant c. The right-hand side assumes its maximum value at
t = log2 x + O(1),
p and a straightforward computation with Stirling’s formula shows that its value
there is O(x/ log2 x). This handles the case of ω.
Hardy and Ramanujan also proved the inequality (5) with Ω in place of ω under the restriction
that
p t ≤ 1.9 log2 x (see [10, Lemma C]). The above argument shows that #{n ≤ x : Ω(n) = t} x/ log2 x for these t. Finally, for t > 1.9 log2 x, the sharper bound #{n ≤ x : Ω(n) = t} x/ log2 x follows from Lemma 7.
It is easy to deduce from Lemmas 7 and 8 that ω and Ω are palindromic only on a set of n of
density zero. It p
is only necessary to observe that the number of palindromes within (log log x)0.51
of log log x is o( log2 x) and then to apply Lemma 8. We leave the details to the reader.
Establishing a corresponding result for the number-of-divisors function d(n) requires somewhat
more intricate arguments.
Theorem 9. Fix an integer g ≥ 2, and assume that g is not a power of 2. For each k, the set of
n for which d(n) is k-nearly-palindromic has upper density Og (g −2k/3 ).
Remark. If g is a power of 2, then the last k digits of d(n) are 0 in base g for almost all n. So
the conclusion of Theorem 9 remains true, but for uninteresting reasons.
We introduce one more piece of notation before embarking on the proof of Theorem 9. For
each natural number n, we let `(n) denote the multiplicative order of 2 modulo n0 , where n0 is the
largest odd divisor of n. Note that for integers h at least as large as the exponent of 2 dividing n,
the residue class of 2h modulo n depends only on the residue class of h modulo `(n).
Proof. The proof borrows some ideas from [4]. Write n = n1 n2 , where n1 is the largest squarefull
divisor of n. Then n2 is squarefree and gcd(n2 , n1 ) = 1. We may assume both of the following
conditions:
(i) n1 ≤ g 4k/3 ,
(ii) |ω(n2 ) − log2
x
n1 |
≤ g k/3
q
log2
x
n1 .
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Indeed, the number of n ≤ x for which (i) fails is O(g −2k/3 x). Now assume that (i) holds. Then
Lemma 7 shows that the number of n2 ≤ x/n1 for which (ii) fails is O(g −2k/3 nx1 ) for large x. (The
notion of large here is allowed to depend on both g and k.) Summing on n1 bounds the total
number of n ≤ x arising in this way by O(g −2k/3 x). So the combined exceptions to (i) or (ii) make
up a set contributing only O(g −2k/3 ) to our upper density bound.
We partition the remaining n into finitely many classes based on the value of the ordered
pair (n1 , ω(n2 ) mod `(g k )). We will show that for each fixed pair of this type, the number of
corresponding n ≤ x is, for large x,
x
g
.
(6)
2k/3
n1 g
`(g k )
Summing over the `(g k ) possibilities for ω(n2 ) mod `(g k ) and then over squarefull n1 ≤ g 4k/3
completes the proof of the theorem.
Given a pair of this type, write the second component of the pair as R mod `(g k ), where 0 ≤
R < `(g k ). Once x is large, (i) and (ii) show that ω(n2 ) is also large. Hence, 2ω(n2 ) is determined
modulo g k by R, and the last k digits of d(n) = d(n1 )2ω(n2 ) are determined by n1 and R. Let B
denote these last k digits.
Since d(n) ≥ 2ω(n2 ) , we see that d(n) > g 2k once x is large. We are assuming that d(n) is
k-nearly-palindromic. So if B̃ is the integer obtained by reversing the digits of B, then B̃ is also
a k-digit integer (i.e., B does not end in zero) and B̃ gives the first k digits of d(n). Choosing the
integer s so that B̃ · g s ≤ d(n) < (B̃ + 1) · g s , we find that
0<
log B̃
log(1 + 1/B̃)
log d(n)
−s−
=
g g −k .
log g
log g
log g
We now look mod 1. Then these inequalities show that loglogd(n)
belongs to an arc An1 ,R of the
g
−k
circle R/Z with length Og (g ).
Write ω(n2 ) = R + t · `(g k ), where t is a nonnegative integer. From (ii), we have t = t0 + t1 ,
where
q


x
k/3
r
−
g
log2 nx1 − R
log
2
n1
x
.
0 ≤ t1 ≤ 2g k/3 `(g k )−1 log2
, and t0 := 


n1
`(g k )


belongs to An1 ,R modulo 1 amounts to the
Since d(n) = d(n1 )2ω(n2 ) , the condition that loglogd(n)
g
requirement on t1 that
log d(n1 )
log 2
k log 2
k log 2
+R
+ t0 `(g )
+ t1 `(g )
∈ An1 ,R .
log g
log g
log g
log g
Since g is not a power of 2, the number `(g k ) ·
log 2
log g
is irrational, and a classical result of Weyl
log 2 ∞
yields the uniform distribution of the sequence {t1 · `(g k ) log
g }t1 =0 modulo 1. As a consequence,
log 2
m
the discrepancy of the sequence {t1 · `(g k ) log
g mod 1}t1 =0 tends to 0 as m → ∞. It follows that
the number of possibilities for t1 , and hence also for ω(n2 ) = R + (t0 + t1 ) · `(g k ), is
r
r
x
1
x
· k = g −2k/3 `(g k )−1 log2
g g k/3 `(g k )−1 log2
,
n1
g
n1
for large x. Lemma 8 shows that the number of values of n2 ≤ x/n1 corresponding to these
possibilities for ω(n2 ) is
r
x
x/n1
x
−2k/3
k −1
g g
`(g )
log2
·q
=
,
2k/3 `(g k )
x
n1
n
g
1
log2 n1
in exact agreement with (6). This completes the proof.
INTEGERS: 14 (2014)
8
3.2. The functions ϕ, σ, and λ, and their compositions
For these functions, we can establish strong results using nothing about palindromes other than
the fact that they are relatively infrequent. Call a set S of natural numbers thin if for all large
x, the number of elements in S not exceeding x is bounded above by x/ exp((log x)c ) for some
constant c = c(S ) > 0. The following theorem was recently established by Vandehey and the
author [12, Theorem 2]:
Proposition 10. Let f be any function of the form f1 ◦ f2 ◦ f3 ◦ · · · ◦ fj , where j is a natural
number, and each fi ∈ {ϕ, σ, λ}. Then f has the property that the inverse image of each thin set
is also thin.
This has the following consequence.
Corollary 11. Let f be any of the functions considered in Proposition 10. Then the set of n for
which f (n) is a palindrome, or is a palindrome after all trailing zeros have been deleted, is a thin
set.
The corollary is immediate from Proposition 10, since the set of integers that are palindromic
in base g after trailing zeros have been removed has counting function Og (x1/2 ).
4. A concluding remark
Perhaps Theorem 1 can also be established using nothing but the sparsity of the set of palindromes.
Indeed, Erdős, Granville, Pomerance, and Spiro conjectured [8, Conjecture 4] that if A is any set of
asymptotic density zero, then s−1 (A ) also has asymptotic density zero. Unfortunately, up to now
nothing nontrivial in this direction has been shown without making further structural assumptions
on A .
References
[1] W. D. Banks, D. N. Hart, and M. Sakata, Almost all palindromes are composite, Math. Res.
Lett. 11 (2004), 853–868.
[2] P. Billingsley, Probability and measure, second ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986.
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